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Unformatted text preview: 18.03 Problem Set 4: Solutions 13. (a) The system is underdamped; the roots are not real. (b) Since the system is underdamped, the solutions are of the form x = Ae bt/ 2 cos( ω d t φ ). Then ˙ x = Ae bt/ 2 (( b/ 2) cos( ω d t φ ) + ω d sin( ω d t φ )). The term following the exponential is sinusoidal, with circular frequency ω d . Therefore it vanishes every π/ω d seconds, and this is the time between a maximum and a minimum, or half of a cycle. We have discovered that π/ω d = 4 / 2 = 2, or ω d = π 2 radians/sec. Since the roots of the characteristic polynomial are b 2 ± ω d i , we have discovered that their imaginary parts are ± π/ 2. (c) For any t , A cos( ω d t φ ) = A cos( ω d ( t +4) φ ), since ω d = π 2 . So the ratio x ( t +4) /x ( t ) is given by the ratio of the other terms: x ( t + 4) /x ( t ) = e b ( t +4) / 2 /e bt/ 2 = e 2 b . This tells us that 1 / 2 = e 2 b , or b = (ln 2) / 2: The real part of the roots is (ln 2) / 4....
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 Spring '09
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 Complex number, general solution

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